500 
MR. G. H. DARWIN ON A PLANET 
§ 4. Graphical solution in the case when there are not more than three satellites. 
Although, a general analytical solution does not seem attainable, yet the equations 
have a geometrical or quasi-geometrical meaning, which makes a complete graphical 
solution possible, at least in the case where there are not more than three satellites. 
To explain this I take the case of two satellites only, and to keep the geometrical 
method in view I change the notation, and write z for E, x for and y for £ 2 , also I 
write n x for fl x , and for fl. 2 . The unit of time is chosen so that A = 1. 
Then the equations (19) become 
dx bz dy bz 
dt = ~bx’ di = ~by . 
and 
2 .( 21 ) 
Now suppose a surface constructed to illustrate (21), x, y, z being the coordinates of 
any point on it. Let the axes of x and y be drawn horizontally, and that of 2 vertically 
upwards. The z ordinate of course gives the energy of the system corresponding to 
any values of x and y which are consistent with the given angular momentum h. 
We have for the dissipativity of the system 
Whence 
( 22 ) 
Let ( X — x)/\=(Y — y)/[jL=Z —2 be the equations to a straight line through a point 
x, y, 2 on the surface. Then if this line lies in the tangent plane at that point 
. "bz . bz 
Kx + %~ 1=0 - 
The inclination of this line to the axis of 2 will be a maximum or minimum when 
X 2 + jh 1 is a maximum or minimum. In other words if this straight line is a tangent 
line to the - steepest path through x, y, 2 on the surface, A-|-/x~ must be a maximum or 
minimum. 
Hence for this condition to be fulfilled we must have 
/xS/x=0 
~ Z S\ + b: 8y = 0 
bx by 1 
And therefore \/r~ = u,/“, and these are both equal to 1 ! 
by / 
bx 
+ 
by) 
